Why can finite elements handle complex geometries, but finite differences can't?

In summary, the advantages of the finite element method over the finite difference method include its ability to handle complex geometries, efficiency in using variational principles, and easy variation of element density. However, not all finite element formulations can handle complex geometries well due to irregular nodal variables and boundary conditions.
  • #1
bumblebee77
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Hello all:

I'm new to the world of finite elements/finite differences. I'd like to understand the advantages of the finite element method. I read that the finite difference method cannot handle complex (e.g., curved domains, fractures) geometries. I have had no luck in understanding why. I would appreciate any help! Thanks.
 
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One reason is that finite differences use "pointwise" approximations, i.e. the derivatives at a particular point in space are approximated from the function values at other points. If the geometry of the points is irregular (expecially in more than one dimension) this is hard to do with high-order accuracy. Efficient finite difference algorithms often use equation-solving methods which rely on a particular pattern of coefficients in the equations, which are are only produced by a regular equally-spaced grid of points.

On the other hand, finite elements can often be formulated using variational principles, which involve minimising a continuous function defined over the area or volume of the element. The function is described by the element's nodal values, and the integration required to minimize the function is done numerically. This is easy to formulate for elements of simple but irregular shapes (e.g. arbitrary shaped triangles or tetrahedra) which can then be used to model any geometrical shape as accurately as required. The density of the mesh can easily be varied to use small elements where the function is changing rapidly and bigger elements elsewhere.

It is not true that all finite element formulations can handle complex geometries well. For example some high-order elements have "weird" nodal variables (for example derivatives like [itex]\partial^3 F / \partial^2 u \partial v[/itex] where u and v are the directions along the edges of the element) which don't match up properly unless adjacent elements have consistent geometry, and/or the corresponding boundary conditions too hard to specify for arbitary shapes.
 
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1. How do finite elements handle complex geometries?

Finite elements use a method called discretization, where a complex geometry is divided into smaller, simpler shapes called elements. These elements are then connected together to approximate the original shape. This allows for a more accurate representation of the geometry compared to finite differences, which use a grid-based approach.

2. What makes finite elements better at handling complex geometries than finite differences?

The use of smaller elements in finite elements allows for a more precise representation of the geometry, as well as the ability to handle curved surfaces and irregular shapes. Additionally, finite elements can incorporate boundary conditions and material properties at the element level, making it a more flexible approach for complex geometries.

3. Why can't finite differences handle complex geometries as well as finite elements?

Finite differences use a grid-based approach that relies on a fixed number of points to approximate the geometry. This can lead to inaccuracies and difficulties in handling curved or irregular shapes. Additionally, finite differences do not have the flexibility to incorporate boundary conditions or material properties at the grid points like finite elements do.

4. Can finite differences be used for any type of geometry?

Finite differences can be used for simple geometries with regular shapes, but they are not well-suited for complex geometries. It is difficult to accurately represent curved or irregular shapes using only a grid-based approach. In these cases, finite elements would be a better choice.

5. Are there any drawbacks to using finite elements for complex geometries?

While finite elements are better suited for complex geometries than finite differences, they require more computational resources and can be more time-consuming to implement. Additionally, the accuracy of the results can depend on the quality of the mesh and the type of elements used. It is important to carefully consider the trade-offs when deciding between finite elements and finite differences for complex geometries.

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