Finite Element Method vs. Integrated Finite Difference for Complex Geometries

In summary, there is a discussion about the best method for modeling flow in a non-rectangular geometry. The question is whether the finite element method is better or worse than the integrated finite difference method. The finite element group claims that the flow field can be distorted with the integrated finite difference method. The main questions are whether this is true and if it is significant, as well as any other potential advantages or disadvantages of each method. The person asking the question has basic knowledge of the methods but needs more information to make a decision. Studiot suggests looking into boundary element methods, which have significant calculation advantages. Recommendations for further reading are also provided. The person is grateful for the help and is looking for objective comparisons of the methods.
  • #1
bzz77
34
0
Hello all:

For modeling flow (or whatever) in a non-rectangular geometry, can anyone comment on whether the finite element method would be better or worse or the same as the integrated finite difference method?

I'm reading some papers by competing groups (so I can decide which code to start using), and the finite element group maintains that the flow field can be distorted when using the integrated finite difference method.

My questions: first of all, is this true? If so, is the problem significant? And are there any other potential advantages/disadvantages of either method over the other?

I have basic knowledge of these methods, but not enough to evaluate their advantages/disadvantages in a meaningful way! Thanks.
 
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  • #2


For fluid modelling, boundary element methods often have significant calculation advantage over either FE or FD methods.

The problem with a FD method is that it is an extrapolation method for which errors can rapidly propagate and grow in the wrong circumstances.


The FE method is an interpolation method which limits this problem

go well
 
Last edited:
  • #3


Thanks Studiot. I'll look into that.

Right now my choice is between integrated finite difference and finite element though. Thanks again.
 
  • #4


For the FD method I can recommend

Numerical Solution of Partial Differential Equations: Finite Difference Methods

GD Smith

Oxford
 
  • #5


Thanks a lot, Studiot.
 
  • #6


Further bibliography

Brebbia has written several books about the boundary element method, including comparisons with FE/FD, working problems both ways.

Boundary elements for engineers

Elements of Computational Hydraulics by

Koutitas

Compares both FD and FE methods with many practical examples.
 
  • #7


Thanks again for going to so much trouble, Studiot. I'll definitely look that up. So far the only comparisons of FE and integrated FD I can find are by the competing groups, and I need something objective. Appreciate it.
 

1. What is the difference between the Finite Element Method and Integrated Finite Difference?

The Finite Element Method (FEM) and Integrated Finite Difference (IFD) are both numerical methods used to solve partial differential equations (PDEs). The main difference between the two methods lies in the way they discretize the domain. FEM divides the domain into smaller, finite elements, while IFD partitions the domain into smaller, finite difference cells. However, both methods follow the same principle of approximating the PDE solution at discrete points within the domain.

2. Which method is better for solving complex geometries?

Both FEM and IFD can handle complex geometries, but FEM is generally more suitable for irregular and non-uniform geometries. This is because FEM allows for more flexibility in the shape and size of the elements, making it easier to accurately represent complex geometries. IFD, on the other hand, requires a more regular and uniform grid, which may be challenging to achieve for complex geometries.

3. What are the advantages of using Finite Element Method?

FEM has several advantages over IFD. Firstly, FEM allows for more accurate representation of complex geometries, as mentioned earlier. Secondly, FEM can handle a wider range of boundary conditions, including non-linear and time-dependent conditions. Lastly, FEM has the ability to adaptively refine the mesh, which can lead to more accurate solutions with less computational effort.

4. When should I use Integrated Finite Difference instead of Finite Element Method?

IFD may be a better choice when the domain is relatively simple and uniform, as the method is easier to implement and requires less computational effort compared to FEM. Additionally, IFD may be more suitable for solving PDEs in one-dimensional or two-dimensional domains, whereas FEM is often used for three-dimensional problems.

5. Are there any limitations to using Finite Element Method or Integrated Finite Difference?

Both FEM and IFD have their own limitations. FEM can be computationally expensive and may require more memory for larger, more complex problems. IFD, on the other hand, may not accurately represent curved boundaries or steep gradients within the domain. Additionally, both methods require careful consideration of the mesh size and element/cell shape to ensure accurate solutions.

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