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mdn
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Please suggest me, how to generate weak form or functional of any partial diffrential equation ( mostely second order) in Finite Element Method.
Thanks in advance.
Thanks in advance.
mdn said:is there any general rule to make weak form, from strong form?
Again i confused about variational formulation, List square and Galerkin method.
timthereaper said:Totally agree with AlephZero. I remember learning the finite element method and it taking months to do, and apparently we'd only scratched the surface. The biggest takeaway from learning FEM for me is that it's a method, not a recipe. It won't tell you exactly how to generate weak forms from strong forms or what approximations to make, but it can tell you what to do next when you have a weak form, for example.
FEM stands for Finite Element Method, which is a numerical technique used to approximate solutions to differential equations in engineering and science. It works by dividing a complex problem into smaller, simpler elements and then solving for the behavior of each element. The solutions from each element are then combined to obtain an overall solution for the entire problem.
A functional is a mathematical expression that maps a set of input functions to a single output value. In FEM, functionals are used to represent the behavior of a system, such as stress or displacement, and are essential in formulating the finite element equations that govern the behavior of each element. They allow us to approximate the behavior of a continuous system using discrete elements, making FEM a powerful tool in solving complex problems.
In order to generate a functional in FEM, you must first define the problem you want to solve, including the boundary conditions and properties of the system. Next, you will need to discretize the problem and create a finite element mesh. Then, using the governing equations and the properties of each element, you can formulate the functional by summing the contributions from each element. Finally, you can use numerical methods, such as the finite difference method, to solve for the unknown variables and obtain the functional.
Some common challenges in generating functionals in FEM include selecting an appropriate element type and mesh density for the problem at hand, dealing with complex geometries, and accurately representing material properties and boundary conditions. Additionally, the process can be time-consuming and computationally demanding, requiring specialized software and computing resources.
FEM has a wide range of applications in science and engineering, including structural analysis, heat transfer, fluid dynamics, and electromagnetics. It is commonly used in the design and analysis of structures, such as buildings, bridges, and aircraft, as well as in the development of new materials and technologies. FEM is also used in medical imaging and modeling biological systems, making it an important tool in many fields of research.