F.E.M and Hamilton's Principle (converting differential equations into integral equations)

In summary, F.E.M (Finite Element Method) and Hamilton's Principle are essential concepts in the field of mechanics and applied mathematics, particularly for solving differential equations. F.E.M involves breaking down complex domains into simpler, finite elements, allowing for numerical solutions of partial differential equations. Hamilton's Principle, which states that the actual path taken by a system is the one that minimizes the action integral, provides a foundational framework for deriving equations of motion. By converting differential equations into integral equations, these methods facilitate the analysis of physical systems, making it easier to apply boundary conditions and solve complex problems in engineering and physics.
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Differential to Integral Equations
Hello

May I begin by saying I do not exactly know what I am asking, but here goes...

In the Finite Element Method (as used in Solid Mechanics), we convert the differential equations of continuum mechanics into integral form. Here, I am thinking of the more pragmatic Principle of Virtual Work, rather that exploiting the more mathematically sophisticated strong/weak formulations (but no matter on that detail)

In Hamilton's Principle, we reformulate Newtonian Dynamics into Analytical Dynamics, but extremizing the Action of the Lagrangian.

Now, in both cases, we convert differential equations into integral equations.

So something is happening here... this act of converting differential into integral. Through the haze of my confusion I can sort of see that the result is more easily addressed with computer programming

Could someone elaborate, perhaps a bit more philosophically, on what is happening when we do these things.

In one sense, both processes relate to variational methods, but is something going on here that these two approaches (sort of) resemble each other, in a way)?

Or am I thinking a bit silly?
 
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FAQ: F.E.M and Hamilton's Principle (converting differential equations into integral equations)

What is the Finite Element Method (FEM)?

The Finite Element Method (FEM) is a computational technique used to approximate solutions to complex differential equations, particularly in engineering and physical sciences. It involves breaking down a large system into smaller, simpler parts called finite elements, which are then solved individually and assembled to form a solution to the original problem.

What is Hamilton's Principle?

Hamilton's Principle, also known as the Principle of Stationary Action, is a variational principle used in classical mechanics. It states that the actual path taken by a physical system between two states is the one for which the action integral is stationary (usually a minimum). This principle provides a powerful method for deriving the equations of motion for a system.

How does FEM convert differential equations into integral equations?

FEM converts differential equations into integral equations through a process called "weak formulation." In this process, the differential equation is multiplied by a test function and integrated over the domain. This integration by parts reduces the order of the derivatives, resulting in an integral equation that can be more easily approximated by finite elements.

Why is Hamilton's Principle important in FEM?

Hamilton's Principle is important in FEM because it provides a systematic way to derive the governing equations of a system in a form that is suitable for numerical approximation. By expressing the problem in terms of an action integral, Hamilton's Principle allows the use of variational methods to obtain the weak form of the equations, which is essential for the finite element discretization.

What are the advantages of using FEM with Hamilton's Principle?

Using FEM with Hamilton's Principle offers several advantages: it ensures that the numerical solution respects the underlying physical principles, it provides a flexible framework for handling complex geometries and boundary conditions, and it allows for systematic improvement of the solution accuracy through mesh refinement and higher-order elements.

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