5

In finite element method, one wants to derive a so-called weak form of the differential equation to solve. This latter is obtained by multiplying both sides of the equation by a "test function" then apply the Green theorem.This test function is written as a sum of shape functions and nodal values, these nodal values can finally be simplified from the equation.

Now in mechanical problems (for instance in linear elasticity), I often read "principle of virtual work", where they apply a "virtual displacement" to the body then equal the external and internal work. The resulting finite element formulation is similar, is this actually the same thing as the general "test function" ?

altprc
  • 51

2 Answers2

3

I realize this question is pretty old, but I'll add my thoughts. The weak form, or variational form, of a PDE is obtained by multiplying the original PDE (or system of PDE's) with a test function(s) and integrating by parts. It turns out for mechanics problems, the principle of virtual work is another name for a particular weak form that is developed with a symmetric functional setting. What do I mean by this? Let us consider the simplest second order equation, say on the interval $(0,1)$: $$u'' = f$$ accompanied with the boundary conditions $u(0) = u(1) = 0$. Multiplying by a test function $v$ and integrating, we arrive at the first weak form of this equation: $$ \int_0^1 u'' v = \int_0^1 fv.$$ Now, we haven't done much to the original PDE, and the energy spaces for $u$ and $v$ are $H^2(0,1)$ and $L^2(0,1)$ respectively. (It is sometimes called the trivial formulation for exactly this reason.) Nevertheless, this is a weak form of the original PDE.

But, we can also integrate by parts. Doing so once gives us another weak form: $$-\int_0^1 u'v' + (u'v)|_0^1 = \int_0^1 fv. $$ Now, the energy setting is $u,v \in H^1(0,1)$. This specific variational formulation is known as the principle of virtual work. Notice that the energy spaces for trial and test functions are the same.

But, we can integrate by parts still! Doing so gives us yet another weak form: $$ \int_0^1 uv'' - (uv')|_0^1+(u'v)|_0^1 = \int_0^1 fv .$$ The energy setting is now $u \in L^2(0,1)$ and $v\in H^2(0,1)$. This formulation, where solution variables are in the most relaxed space, is sometimes known as the ultraweak variational formulation, since it puts all the regularity on the test functions.

The take home message is this: for a given PDE, we may be able to develop several weak (or variational) formulations that result in different functional settings. One particular formulation, the one in which the functional setting is symmetric, is commonly referred to as the principle of virtual work, which, unfortunately, is the only one commonly taught to engineers.

Jonathan Zhang
  • 41
2

To the best of my knowledge, both the weak form and virtual work are identical. At least in the mechanic's context. However, the concept of weak form is general and applies to non-mechanical systems too. The idea behind the weak form is-- instead of finding a solution to the differential equations (strong form), multiply the differential with a test function and find a solution over the complete domain. Since the test function is arbitrary, so solutions to both weak form and differential equations are identical. In the finite element method, we replace the test functions with the basis functions.

akr
  • 45