Finite functions (Ограниченная функция – отлична от нуля только в пределах треугольника)
Finite function inside the triangle may have different dependences. Everything depends on the value of the nodal potential.
There are three very special finite function:
F
inite
function associated with the first node:
Similar relations are valid for 2 other finite functions.
Finite functions for triangles = simplex coordinates (coordinates – because every point inside the triangle may be describe by certain value of finite functions, that is why the behavior of these functions is similar to coordinates).
Simplex coordinates
Two-dimensional simplex coordinates:
Another definition of the point position is a set of simplex coordinates
Really only 2 of 3 simplex coordinates are independent:
General relation for the simplex coordinates:
Approximation of functions inside triangles (Аппроксимация функций внутри треугольника)
Approximated function is the potential: U(x, y)
Electric
field intensity
Potential
inside a triangle (Wave sign
means ‘approximation’):
Finite function:
Approximation of the field intensity:
Approximation of the equation (Аппроксимация уравнения)
Potential inside a triangle (Wave sign means ‘approximation’):
Finite function:
Equation:
If we substitute approximation of the potential into this equation we will lose information completely, because
Weighted residual method (метод взвешенных невязок)
This method is used to solve different kinds of problems and the idea behind this method is non-trivial. Let’s look to the initial equation:
Let’s choose some arbitrary (произвольную) function ψ(x,y) (можем выбрать некоторую произвольную функцию, которую в будущем назовём взвешенной (weighting) функцией). Evidently, if right initial equation is equal to zero, so the product of this Laplace operator and weighting function will be equal to zero.
Now let's integrate this product over the area s (this s corresponds to the whole problem domain)
Also, it is necessary to remember, we are talking about finite element method and all finite functions and consequently these weighted functions also will be limited by certain element, they will be equal zero outside the considered element and will be non-zero inside them. That’s why the first glance (взгляд) this integral is very complicated it should be integrate over the wide area, but in the practice it should integrate always in very small area, which is restricted by considered elements.
Galerkin method (метод Бубнова-Галеркина)
We can use many different weighting functions. The math tells us if we shall take big number of different functions and if we shall insure this equation for different weighting functions, then we shall get a good approximation of the initial potential distribution. It will be not exactly the potential distribution, it will be something, which is not very far from this distribution. Moreover, the final approximate solution will depend on how we shall choose this weighting function. For different set of the weighting functions, we shall get different solutions and all of them may be called as approximations of the solution.
So, how to choose them, what to do. The answer is done in the Galerkin method.
This is an initial equation and that is true only because the Laplacian of potential is equal to zero. Evidently, the Laplacian of such approximation not necessary will be equal to zero it may differ from zero, that’s why the integral from the product of Laplacian weighting function and after integration will give us so called ‘weighted residual’ – R:
So, the idea of the Galerkin method – it is the special choice of the weighting functions:
to use the weighted residual method;
to use approximation functions for weighting:
to set residuals to zero; Rj=0
to apply integration-by-parts procedure to the integral (weak formulation)
Let’s assume that the weighting function is the same of the finite function, which corresponds just to this triangle, of course, there are three different functions, so for every triangle we should use three weighting functions. The second assumption, set residuals to zero, evidently we can choose approximation of the potential in such a way the residual will be really equal to zero, not because the Laplacian of the potential is equal to zero, but because the product of this potential and the weighting function after integration will give zero. And now let’s apply integration-by-parts procedure to the integral.
Indeed, if we shall directly calculate this integral using approximation of the potential, we can get exact zero (zero will be equal to zero), so to avoid this difficulty it is possible to use some mathematic transformation, which is called integration-by-part. Очень важная часть метода Галеркина заключается в том, что прямолинейное вычисление интеграла такого вида даст нам нулевой ответ только потому, что вторая производная от полинома первого порядка всегда равна нулю, и таким образом мы не сможем найти решение. Для того, чтобы преодолеть эту сложность в методе Галеркина предполагается провести идентичное преобразование – интегрирование по частям.