For traditional FEM and FVM, why can't we use mesh to represent geometry and use the mesh which represent the geometry to do the computation directly?

Question

Isogeometric analysis [1] has the advantage of integrating geometric and mesh models using NURBS or Spline. At the same time, I would like to ask a question to my friends: for traditional FEM and FVM, why can't we use mesh to represent geometry and use the mesh which represents the geometry to do the computation directly?

In detail, according to the discrete differential geometry, if we can find the equivalent definitions between continuous and discrete geometry, we can represent a continuous geometry with a mesh. We know that for CG software (for example Blender), the main way of modelling is to use the mesh to represent the continuous geometry, we modify the geometry by modifying the mesh. But for the scientific computing mesh generation software (for example Gmsh), we usually need a geometric model first and then set some parameters to do the meshing. There is a big difference between them. I know CG mainly uses the surface mesh in 3D space. And scientific calculation uses mainly solid mesh. But I think that CG's way of using mesh to represent geometry is simpler and suitable for shape change (optimization). So I wonder why can't we use mesh to represent geometry like CG in scientific computation. The geometry is directly represented by a mesh, and then you can do the computation directly on it. I think for the pioneers of FEM and FVM, the idea of combining the geometry and mesh must have been considered before. But now almost all the simulation software don't combine the geometry and mesh model. So I wonder what is the main difficulty of not doing that. So I would like to ask the question to my friends: for traditional FEM and FVM, why can't we use mesh to represent geometry and use the mesh which represents the geometry to do the computation directly?

This question is inspired by the answers of my previous question: Can the mesh generation methods in FVM and FEM be totally based on the knowledge of the mesh generation theory in computer graphics?

bibliography

1. Hughes, T. J. R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng. 194, No. 39-41, 4135-4195 (2005). ZBL1151.74419.

Answer 1

The underlying problem is that the mesh really isn't the geometry. You want to simulate a bridge? It has a certain geometry, which you can approximate using a mesh, but the mesh is not the exact geometry. The same is true for most other curved objects.

There are of course approaches to integrate the geometry into the finite element mesh. In particular, I'd like to point you to this preprint. The primary obstacle is historical: the traditional workflow is to describe the exact geometry using CAD software; this CAD geometry is fed to the mesh generator that creates a mesh; the mesh is then given to the finite element ("analysis") software, but it no longer has access to the CAD geometry. The preprint linked to has an extensive description of what one needs to do to route the CAD geometry all the way through to the analysis engine, and why that is useful.

Answer 2

"So I wonder why can't we use mesh to represent geometry like CG in scientific computation. The geometry is directly represented by a mesh, and then you can do the computation directly on it." ...

This is exactly what we do using iso-parametric elements in FEM; we discretise the geometry and field variables using the same mesh. The concept of iso-parametric elements is not limited to the Lagrange family of elements or NURBS. In theory, one can use any appropriate polynomial space for discretising the geometry and field variables, for example, subdivision surfaces, T-Splines, Box-splines, Chebyshev polynomials etc.

In FVM also the same mesh is used for both geometry and field variables. But the concept of iso-parametric representation is not applicable to FVM, especially for cell-centred FVM approaches. Please correct me if I am wrong.

IGA sounds very promising at the first-look. It certainly has it's advantages when compared to the traditional FEM. IGA is already available in LS-DYNA. But, I think that IGA's disadvantages at the moment outweigh it's advantages when it comes to practical applications. There are also other issues that are not technical, see point 5.

1.) IGA is not free from the discretisation of the space. One still has to discretise the geometry (to differentiate it from the word "mesh", if that makes some sense). Talking in terms of IGA, one needs to generate a new knot-space for NURBS. Such a feature is not yet available in CAD software. It is mostly carried out either manually or using custom scripts in academic research groups.

2.) IGA is rooted in the idea of higher continuity across element boundaries. While higher continuities are advantageous in FEM for some specific problems especially for problems with smooth solutions, this point is not so advantageous for problems with discontinuities in the field variables. Moreover, higher continuities are limited to an individual patch without sharp corners.

Furthermore, discontinuities across elements are inherent to the cell-centred FVM. Therefore, I don't think that it makes sense to use IGA directly for FVM. (I don't have much experience in FVM to elaborate further on this).

3.) IGA is expensive when compared with the traditional FEM. Higher continuity means more non-zero entries in the (effective) stiffness matrix. While it is true that we can get accurate results using coarse meshes with higher-order elements, it doesn't pay off much to use higher than quadratic polynomials for practical applications. In my experience, quadratic polynomials are the optimum choice when taking accuracy and efficiency (runtime) into consideration.

4.) It is also difficult to extend IGA to advanced problems in solid mechanics, for example, plasticity and incompressible hyperelasticity, which require sophisticated formulations.

• Projection-based methods, for example, F-bar (or B-bar as some call it) formulation, require the inversion of the patch-wise mass matrix. -> Expensive.
• Taylor-Hood type elements (polynomial for pressure field is one order lower than that for displacement (or velocity) field) are not inf-sup stable. One needs to use sub-division stabilisation to ensure inf-sup stability. (similar to the Q1-iso-Q1 element). This has practical limitations.

It would have been fantastic if we didn't have to resort to the sophisticated formulations when using IGA. But, unfortunately, we still have to use those advanced formulations.

5.) Design and analysis workflows in the industry are already well-established based on the software tools that use traditional FEM and FVM. It is extremely difficult to convince industries to change their workflows to use new simulation paradigms unless we show them some quick rewards or significant savings.

It is safe to say that IGA is not matured enough to replace traditional FEM. It is hard to say when it will. But one can combine the concepts and tools from IGA and traditional FEM to improve upon the existing schemes, see paper1 and paper2.

Answer 3

Yes, you can use the same mesh to represent the geometry of your domain and to solve the PDE, that is something that you can do. For example, a square can be completely described by two triangles and you can use this mesh to solve your PDE. If the solution of your PDE is constant or linear this mesh would suffice but if you have higher gradients in your solution you would need more elements.

In general, a sequence of meshes is considered and there is a fine-enough mesh (for your application) that gives you an error under your desired tolerance. For example, solving

$$\nabla^2 u = -80 \sin(4x) \cos(8y)\, ,$$

for $(x, y) \in [0, \pi]^2$ requires a mesh that is a square. Nevertheless, using just two element would not give you a solution close to the analytic result $u = \sin(4x) \cos(8y)$.