What Does Variational Formulation Mean in Finite Element Methods?

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In summary, the variational formulation is a mathematical approach to solving problems in physics and engineering, where the problem is reformulated as an optimization problem and the solution is found by minimizing a functional. This approach is more elegant and efficient compared to the traditional formulation based on differential equations. A functional in this context is a mathematical function that maps a set of functions to real numbers, representing the objective to be minimized. Boundary conditions are crucial in variational formulation as they define the solution space and ensure physical constraints are satisfied. The Euler-Lagrange equation is used to find the necessary conditions for the unknown functions to minimize the functional and obtain the solution. Variational formulation has a wide range of applications in fields such as classical mechanics, electromagnetics,
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pamparana
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Hello,

See this word a lot on internet articles about FEM and such. However, no one explains what actually the term means? I could see that it has something to do with starting with approximate solutions.

Anyone has any idea about this?

Thanks,
Luc
 
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Finite Element Methods: they are used to approximate solutions of PDE's.
 

FAQ: What Does Variational Formulation Mean in Finite Element Methods?

What is the difference between the variational formulation and the traditional formulation of a problem?

The variational formulation is a mathematical approach to solving problems in physics and engineering, while the traditional formulation is based on differential equations. In the variational approach, the problem is reformulated as an optimization problem, where the solution is found by minimizing a functional. This often leads to a more elegant and efficient solution compared to the traditional approach.

What is a functional in the context of variational formulation?

A functional is a mathematical function that maps a set of functions to real numbers. In variational formulation, the functional represents the objective to be minimized in order to find the solution to a problem. It is typically expressed as an integral of a Lagrangian function, which depends on the unknown functions and their derivatives.

What is the importance of boundary conditions in variational formulation?

Boundary conditions are essential in variational formulation because they define the set of functions that the optimization process will consider. They are used to constrain the solution space and ensure that the solution satisfies the physical constraints of the problem. Without proper boundary conditions, the solution obtained may not be physically meaningful.

How is the Euler-Lagrange equation used in variational formulation?

The Euler-Lagrange equation is a necessary condition for the solution of a variational problem. It is derived from the functional and represents the critical points where the functional is stationary. In other words, it gives the necessary conditions for the unknown functions to satisfy in order to minimize the functional and obtain the solution to the problem.

What are some common applications of variational formulation in science and engineering?

Variational formulation has a wide range of applications in various fields, such as classical mechanics, electromagnetics, fluid dynamics, and quantum mechanics. It is also commonly used in optimization, control theory, and numerical analysis. Some specific examples include the calculus of variations, finite element analysis, and the time-dependent Schrödinger equation.

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