Methods for determining residuals. Weighted residual method. Field theory problems

UNESCO Chair for NIT, Rein T.S. Introduction The function is approximated by a set of functions: Where - unknown parameters - linearly independent functions belonging to the complete sequence (3) Consider the error function (residual): (4) In this case, we will assume that: - set of weight functions (5)

UNESCO Chair for NIT, Rein T.S. Collocation method. Example Consider the following second-order equation on the interval: with boundary conditions: Let us take the approximating function in the form of an expression that satisfies the boundary conditions for any: (6) (7) (8) for Exact solution (check): We choose as collocation points

UNESCO Chair for NIT, Rein T.S. Collocation method and least squares method Let us extend the collocation method to the case when the number of points exceeds the number of unknowns. In this case, the unknown parameters are determined by minimization in the root mean square sense. is estimated at points (), and the function can be written in the form: We minimize (16), for the th equation we obtain: (15) (16) (17)

UNESCO Chair for NIT, Rein T.S. Example Consider the following second-order equation on the interval: with boundary conditions: and an approximating function in the form of an expression that satisfies the boundary conditions for any: for Exact solution (check): Calculate the discrepancy at three points:

UNESCO Chair for NIT, Rein T.S. Method of moments For a given system of equations: Any set of linearly independent functions from the complete sequence can be used as weight functions, for example: This ensures that the higher order residual moments vanish: (18) (17) (19)

UNESCO Chair for NIT, Rein T.S. Example Consider the following second-order equation on the interval: with boundary conditions: and an approximating function in the form of an expression satisfying the boundary conditions for any: for Exact solution (check): The error function is orthogonalized with respect to and:

Having studied one method in relatively detail, we move on to presenting other methods in entire classes. The most common class is weighted residual methods. They proceed from the assumption that the desired function can be represented in the form of a functional series, for example this:

They usually try to choose the function f 0 so that it satisfies the initial and boundary conditions as accurately as possible. The approximating (test) functions f j are assumed to be known. Mathematicians have come up with a number of requirements for such functions, but we will not discuss them here. Let us confine ourselves to the fact that polynomials and trigonometric functions satisfy these requirements. Several more examples of sets of similar functions will be considered when describing specific methods.

The coefficients a j are unknown in advance, and they should be determined from a system of equations obtained from the original equation. From an infinite series, only a certain finite number of terms are taken.

In the equation that is supposed to be solved, all terms are rewritten to the left side, leaving only zero on the right side. Thus, the equation is reduced to the form

If an approximate solution (written as a finite sum of pre-selected functions) is substituted into this equation, then it will not be identically satisfied. Therefore, we can write

where the value R is called the residual. In general, the residual is a function of x, y, z and t. The problem comes down to finding such coefficients a j that the discrepancy remains small throughout the entire computational domain. The concept of “small” in these methods means that the integrals over the computational domain of the residual multiplied by some weighting functions are equal to zero. That is

Having specified a finite number of weight functions, we obtain a system of equations for finding unknown coefficients. By specifying various trial approximating (trial) and various weighting functions, we easily obtain a whole class of methods called weighted residual methods.

Here are some examples of the simplest methods from this class.

Subarea method. The computational domain is divided into several subdomains D m that can overlap each other. The weight function is specified in the form

This ensures that the integral of the residual over each subdomain is equal to zero. The method served as the basis for a number of methods (one of them will be discussed below).

Colocation method. The Dirac delta function is used as weight functions

Where x=(x,y,z). Let me remind you that the Dirac function is a tricky function that is equal to zero everywhere except the origin. But at the beginning it takes on a value unknown to science such that any integral over the region containing the origin of coordinates is equal to unity. To put it simply: we set a certain number of points (often called nodes in this approach). The original equation will be satisfied at these points. There are approaches to selecting these points and trial functions to maximize accuracy with a limited number of nodes. But we won’t discuss them here.

Least square method. The method is based on minimizing the value

But it is not difficult to show that it also belongs to the class of weighted residual methods. The weight functions for it are functions of the form

Perhaps this is the most famous method from this class among non-specialists, but it is far from the most popular among specialists.

Galerkin method. In this method, approximating (trial) functions are taken as weight functions. That is

The method is widely used in cases where one wants to find a solution in the form of a continuous (rather than a grid) function.

Let us consider the application of these methods to the calculation of the deformation of a cantilevered beam of length L. Let the deviation from the center line be described by the equation

Boundary conditions are specified in the form

We will look for a solution in the form

Then the discrepancy will be written in the form

To find the unknown coefficients a and b, we will need to create a system of two equations. Let's do this using all the methods discussed.

Colocation method. Select two points at the ends of the beam. We equate the discrepancy in them to zero

We get

As you can see, the collocation method is quite simple to implement, but is inferior in accuracy to other methods.

Subarea method. We divide the entire length of the beam into two subregions. In each of them, we equate the integral of the residual to zero.

Galerkin method. We take the integrals of the residual multiplied by the test functions.

Least square method.

The least squares method requires the greatest computational effort, but does not provide a noticeable gain in accuracy. Therefore, it is rarely used in solving practical problems.

50. EXPLICIT AND IMPLICIT DIFFERENCE SCHEMES. WEIGHTED RESIDENTS METHOD. BUBNOV-GALERKIN METHOD.

Difference scheme- this is a finite system of algebraic equations, put in correspondence with some 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, which has a continual nature, to a finite system of equations, the numerical solution of which is fundamentally possible on computers. Algebraic equations put into correspondence 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).

The solution of the difference scheme is called an approximate solution of the differential problem.

Although the formal definition does not impose significant restrictions on the type 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 schemes calculate the value of the result through several neighboring data points. Example of an explicit scheme for differentiation: (2nd order approximation). Explicit schemes often turn out to be unstable.

Here V * – approximate solution,
F– function that satisfies the boundary conditions,
N m – test functions, which must be equal to zero at the boundary of the region,
A m – unknown coefficients that need to be found from the condition of best satisfaction of the differential operator,
M– number of trial functions.

If we substitute V* into the original differential operator, we obtain a discrepancy 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 different variants of the weighted residuals method are distinguished,

S– region of space in which a solution is sought.

When choosing delta functions as weight functions, we will have a method called pointwise collocation method, for piecewise constant functions - the method of collocation by subdomains, but the most common is the Galerkin method, in which test functions are selected 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 matrix K and vector Q are calculated using the formulas:

After finding the coefficients A and substituting them into (1), we obtain a solution to the original problem.

The disadvantages of the weighted residuals method are obvious: since the solution is sought over the entire domain at once, the number of trial and weight functions must be significant to ensure acceptable accuracy, but this creates difficulties in calculating the coefficients K ij And Q i, especially when solving plane and volumetric problems, when it is necessary to calculate double and triple integrals over areas with curvilinear boundaries. Therefore, this method was not used in practice until the finite element method (FEM) was invented.

PROBLEMS OF FIELD THEORY

The finite element method is a numerical method and is based on replacing an object (structure or part thereof) with a set of subdomains (elements), for each of which an approximate solution to the heat transfer problem is found. This means that for each element it is necessary to write down the differential transport equation and boundary conditions characterizing the heat transfer processes on the boundary surfaces of this particular element, and then obtain a solution in one form or another. Combining “elemental” solutions according to a certain rule provides a solution to the problem for the object as a whole. This chapter will introduce the basic concept of FEM.

2.1 Weighted residual methods

A large group of methods for approximate solution of differential

equations is based on a mathematical formulation related to

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 for it:

,
, (2.1.1)

,
. (2.1.2)

Here L−differential operator; x i− spatial coordinates; V And S− volume and outer boundary of the study area; u 0 - exact solution.

We will assume that some function u is also a solution to the equation, and it can be approximated by a set of functions
:

, (2.1.3)

while the coefficients − unknown quantities that must be determined using some mathematical procedure.

In residual methods, this procedure consists of two successive steps. At the first stage, by substituting the approximate solution (2.1.3) into equation (2.1.1), we find the function
error, or residual, which characterizes degree of difference
from accurate solutions :

The result is an algebraic equation 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 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 . In these M points, the discrepancy must be equal to zero, which leads to the system M algebraic equations for M coefficients :

. (2.1.5)

In weighted residual methods, a weighted residual is first formed by multiplying it by some weighting functions , and then minimize it on average:

. (2.1.6)

In the least squares method - the Rayleigh-Ritz method - the error itself is chosen as the weight function, i.e.
, and it is required that the value (functional) obtained in this way be minimal:

. (2.1.7)

To do this, the following condition must be met:

, (2.1.8)

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 they are required orthogonality to residual :

. (2.1.9)

If is a linear operator, then system (2.1.9) goes over to a system of algebraic equations for the coefficients .

Let's consider the Galerkin method using a specific example. Given an equation on the interval
:

with boundary conditions:
,
.

Let's take the approximating function in the following form:

satisfying boundary conditions (2.1.2) for any . At the first stage we find the discrepancy:

Let's perform the procedure of the second stage:

,
.

Integration will lead to a system of two equations:

,

the solution of which will be the following values :
;
. An approximate solution has the form:.

A comparison of approximate results obtained by various methods with the exact solution is given in Table 1.

Table 1

From Table 1 it is clear that with the same approximating functions in all methods, the best approximation to the exact solution is provided by the Galerkin method. In addition, this method is applicable to solving nonlinear problems, including those for which there is no functional required when using the Rayleigh-Ritz method.