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Meaning of Weak form use in Finite element Method

rohan_patel8388's picture



 I am a Computational Mechanics Student. I always come across during class of Finite Element Methods. I want to know what does actual mean by "Weak Form"? IAnd What are we take Test Function while using in Galerkin Method?



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Weak form means, instead of solving a differential equation of the underlying problem, an integral function is solved. The integral function implicitly contains the differential equations, however it's a lot easier to solve an integral function than to solve a differential function. Also, the differential equation of system poses conditions that must be satisfied by the solution (hence called STRONG form), whereas, the integral equation states that those conditions need to be satisfied in an average sense (hence WEAK form). However, do not understimate the power of integral functions just by its name "weak form".

An example is stresses on free surface. Strong form says if there is no traction on the surface, corresponding stresses must be zero.So if we can solve the PDE, we'll get exact zero stresses. But all FEA code will give you some stress on the surface since whole FEA is based on integral functions. So as you refine the mesh, the stress value at traction free surface goes closer and closer to zero...this is average sense :)


The world started with 0, is progressing with 0, but doesn't want 0.

For more understanding on these, read these books:

 Chapter 3: Fundamentals of Finite Element Method (Vol I) by ZienKiewicz & Taylor -> very elaborate

Chapter 3: Concepts and Application of Finite Element Analysis, by Cook ->short but the point :)



The world started with 0, is progressing with 0, but doesn't want 0.

rohan_patel8388's picture

Thnaks for reply. I will study it.

mohamedlamine's picture

A Test Function in Weighted Residuals is not the Exact Solution It is an Approximation of the Exact and is Intuitivly assumed.

In some cases it can Fit the Exact. With Galekine Method it Leads to the Finite Element Formulation.



Junwei Xing's picture

strong form: PDE form.

weak form: integration form, which can be obtained by weighted integration of the PDE and then using integration by part. 

why PDE is "strong" form and integration form is "weak form"? because for example the PDE is 1-d Poisson, i.e.  -u''=f. then to make sense of the term -u'' in traditional sense, function u should belong to C2 function space, while f belongs to c0. the integration counterpart of -u''=f is int(u'*v'-f*v)=0, to make sense of the integration term in(u'*v') we need select functions u and v from function space H1. to make sense of int(f*v) we need to select f from L2, or more larger space H^-1. Because the function space H1 required for u in integration form is larger than function space L2 required for u in PDE form, so we say the integration form is weak form and PDE form is strong form. Actually, the integration form relaxes the regularity (smoothness) requirement of the unknown function u, which is the reason it's called weak form. In the other hand, the regularity requirement for u in PDE is too strong for real world engineering problems, such as problems with sharp geometry change, the integration form is more suitable for real world engineering problems.


The above is my understanding the definition of "strong form" against "weak form". 

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