Calculation boundary terms of a functional

In summary, The problem is to derive equilibrium equations in u, v, and w directions from which boundary terms must be found. The author thinks they have derived the equations, but doesn't know how to proceed.
  • #1
Dear all,

I am stuck with the problem which is given below;
attachment.php?attachmentid=62140&stc=1&d=1380043019.png


In this problem the equilibrium equations of the given functional must be derived in u, v, and w directions from which the boundary terms must be found. I think that i have derived the equilibrium equations( 5 equations), but i don't know how to proceed. Does anyone maybe know how to do it??


Thanks in advance,
 

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  • #2
ronaldinho52 said:
Dear all,

I am stuck with the problem which is given below;
attachment.php?attachmentid=62140&stc=1&d=1380043019.png


In this problem the equilibrium equations of the given functional must be derived in u, v, and w directions from which the boundary terms must be found. I think that i have derived the equilibrium equations( 5 equations), but i don't know how to proceed. Does anyone maybe know how to do it??

I think you're missing a name for the integrand. I will use [itex]L[/itex].

To determine [itex]\delta I[/itex], integrate [itex]L(u + h, \dots, v + k, \dots, w + l, \dots) - L(u, \dots, v, \dots, w, \dots)[/itex] term by term. You can swap the order of integration so that terms involving derivatives with respect to x are integrated with respect to x first, and terms involving derivatives with respect to y are integrated with respect to y first.

For example, for the [itex]u_{,x}[/itex] term you get:
[tex]
\int_a^b \int_c^d h_{,x} \frac{\partial L}{\partial u_{,x}}\,\mathrm{d}x \,\mathrm{d}y
= \int_a^b \left[ h \frac{\partial L}{\partial u_{,x}}\right]_c^d\,\mathrm{d}y - \int_a^b \int_c^d h \frac{\partial}{\partial x} \left(\frac{\partial L}{\partial u_{,x}}\right)\,\mathrm{d}x\,\mathrm{d}y[/tex]

(The antiderivative with respect to x of [itex]h_{,x}[/itex] is [itex]h + A(y)[/itex] for an arbitrary function [itex]A[/itex]; but it will be the same function at both [itex]x = c[/itex] and [itex]x = d[/itex], so it cancels out.)

Ideally the conditions of your problem are such that the perturbation [itex]h[/itex] vanishes everywhere on the boundary, so the boundary term vanishes.

With the second derivatives of w you have to integrate by parts twice:
[tex]
\int_a^b \int_c^d l_{,xx} \frac{\partial L}{\partial w_{,xx}}\,\mathrm{d}x \,\mathrm{d}y
= \int_a^b \left[ l_{,x} \frac{\partial L}{\partial w_{,xx}} \right]_c^d\,\mathrm{d}y
- \int_a^b \int_c^d l_{,x} \frac{\partial}{\partial x}\left(\frac{\partial L}{\partial w_{,xx}}\right)\,\mathrm{d}x \,\mathrm{d}y
[/tex]
Again, one would hope that the conditions of your problem require that [itex]l_{,x}[/itex] vanishes on the boundary, so the boundary term vanishes. The remaining term is dealt with as for the other first derivatives.

Incidentally, I believe there should be three equilibrium equations (one each for u, v and w) so if you have five then you have made an error somewhere.
 
  • #3
I will try to solve the question now with your explanation. Thank you very much!
 
  • #4
Regarding your explanation I ended up with the following equation to be solved.

attachment.php?attachmentid=62162&stc=1&d=1380059658.jpg


Is this correct or did I get your explanation totally wrong??(btw i forgot the dxdy term at the end of the equation)
 

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What is a functional?

A functional is a mathematical concept that takes a function as its input and produces a numerical value as its output. It can be thought of as a "function of a function".

What are calculation boundary terms?

Calculation boundary terms refer to the values at the boundaries of a function's domain. These values can affect the overall value of a functional and must be taken into account when calculating it.

Why is it important to calculate boundary terms of a functional?

Boundary terms can significantly impact the value of a functional and must be considered in order to accurately describe the behavior of a system or function.

How do you calculate boundary terms of a functional?

The exact method for calculating boundary terms can vary depending on the specific functional and its domain. In general, it involves evaluating the function at the boundaries and incorporating these values into the overall calculation.

What are some real-world applications of calculating boundary terms of a functional?

Calculating boundary terms of a functional is important in many scientific fields, such as physics, engineering, and economics. It can be used to model and analyze the behavior of complex systems, predict outcomes, and make informed decisions based on quantitative data.

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